1. Field of the Invention
The present invention relates to a method for simulating the distribution of impurities that has been implanted in a semiconductor device manufacturing process and particularly, to an ion-implantation simulating method in which dose is conserved.
2. Description of the Prior Art
In a semiconductor manufacturing field, a method for simulating the distribution of impurities with a process simulator has been used to estimate the characteristics and operation of devices in place of experimental or tentative measurements of the characteristics of devices. A method for simulating a two-dimensional profile of implanted ions using analytic equations is explained in "Depth Profiles of Boron Atoms with Large Tilt-Angle Implantations", J. Electrochem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY, pp. 996-998 (1986). According to this publication, an distribution of implanted ions C.sub.2d (x,y) for the two-dimensional shape is expressed by the following equation (1): ##EQU1## Here, C.sub.i (y) represents a distribution of implanted ions in one-dimensional structure which is calculated for an i-th layer, .sigma..sub.l represents a standard deviation of a re-distribution in the transverse direction for calculating a two-dimensional distribution of implanted ions based on a one-dimensional distribution of implanted ions, and erfc represents a complementary error function.
A method of calculating a one-dimensional impurity distribution C.sub.i (y) is explained in "Analysis and Simulation of Semiconductor Devices", Springer-Verlag Vien New York) as follows.
The distribution of implanted ions in one-dimensional single layer structure can be expressed by three kinds of analytic equations, i.e., the equation of a Gaussian distribution, the equation of joined Gaussian distribution and the equation of a Pearson distribution, which use parameters, Rp, .sigma., .gamma., .beta., called moments and characterizing the distribution of impurities after ion-implantation. Rp represents a range which indicates the average depth of the impurity distribution after the ion-implantation, .sigma. represents a deviation which indicates the breadth of the impurity distribution after the ion-implantation, .lambda. represents a skewness which indicates the distortion of the distribution, and .beta. represents a kurtosis indicating the sharpness of the distribution, and these parameters are extracted from profiles or the like which were actually measured in advance. These moments are determined in accordance with the combination of implantation energy, dose, substrate material, ion type, etc.
Here, for example, the Pearson distribution will be explained as follows. A function I(y) representing the Pearson distribution is given by the following equations using Rp, .sigma., .gamma., .beta.: EQU dI(y')/dy'=(y'-a)I(y')/(b.sub.0 +ay'+b.sub.2 y'.sup.2) (2) EQU y'=y-Rp (3) EQU a=-.sigma..gamma.(.beta.+3)/A (4) EQU b.sub.0 =-.sigma..sup.2 .gamma.(4.beta.-3.gamma..sup.2)/A (5) EQU b.sub.2 =(-2.beta.+3.gamma..sup.2 +6)/A (6) EQU A=10.beta.-12.gamma..sup.2 -18 (7)
Function I(y) is normalized so that the integral value thereof is equal to 1, and the dose specified at the time of the ion-implantation is multiplied to function I(y) to calculate the distribution after the ion-implantation.
In place of the above three kinds of equations, a dual Pearson distribution obtained by summing up two independent Pearson distributions may be used.
A method for simulating the distribution implanted ions in one-dimensional multilayer structure can be obtained by expanding the method for distribution implanted ions in one-dimensional two-layer structure which is described in "Models for Implantation into Multilayer Targets", Appl. Phys. A41, pp.201-207 (1986) (by H. Ryssel, J. Lorens, and K. Hoffmann). The material is defined for each layer of the substrate, and the moment for each material is given in advance. Therefore, the impurity distribution I(y) which is normalized so that the dose is equal to one can be calculated for each layer. In order to simulate the distribution of implanted ions in the multilayer structure, the uppermost layer is set as a first layer, the normalized impurity distribution for a k-th layer is defined by equation (8) and the actual impurity distribution is defined by equation (9): ##EQU2## where t.sub.i represents the thickness of i-th layer, and R.sub.pi represents the range in i-th layer. R.sub.pk represents the range in the layer for which the distributions are calculated (i.e., the k-th layer). D.sub.k represents the dose of k-th layer, and is obtained by subtracting from the total dose a partial dose which has been consumed until reaching the (k-1)-th layer.
The impurity distribution which has been calculated on the basis of analytic equation (1) is then passed to a simulator for oxidation, diffusion, etc. According to "Simulator for Semiconductor Device Design" written by Hideteru Koike, et al. and edited by Fuji Sogo Research, pp114-115, the impurity distribution which is handled by the simulator for oxidation, diffusion, etc. is required to be defined on a differential mesh shown in FIG. 6. In FIG. 6, ai represents the width of the differential mesh along x direction, bi represents the width of the differential mesh along y direction, hi represents the distance between the center points of the adjoining meshes along x direction, and ki represents the distance between the center points of the adjoining meshes. One differential mesh is referred to as "control volume".
The impurity concentration which is scalar quantity is defined, for example, at the center point of the different mesh. Therefore, an impurity distribution C.sub.2d ((x.sub.i +x.sub.i+1)/2, (y.sub.j +y.sub.j+1)/2) at the coordinate ((x.sub.i +x.sub.i+1)/2, (y.sub.j +y.sub.j+1)/2) at which the scalar quantity of FIG. 5 is defined is calculated from equation (1), and is defined on ((x.sub.i +x.sub.i+1)/2, (y.sub.j +y.sub.j+1)/2) of FIG. 5.
In the above-described conventional technique, the impurity concentration on a cell (control volume) which is cut out by the meshes is represented by the impurity concentration at the center point of the cell which is given by the equation (1). However, the concentration of the center point does not accurately reflect the impurity concentration of the overall cell. Therefore, even when the impurity concentration is integrated over all the areas by using the impurity concentrations determined as described above to calculate the dose, the dose thus calculated is not equal to the preset dose. Accordingly, it was impossible to accurately estimate the characteristics of a device even when the characteristics of the device is estimated on the basis of the impurity concentration thus determined.